Show commands:
SageMath
E = EllipticCurve("hc1")
E.isogeny_class()
Elliptic curves in class 47040.hc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.hc1 | 47040dl4 | \([0, 1, 0, -16464065, -25718546337]\) | \(128025588102048008/7875\) | \(30359089152000\) | \([2]\) | \(1179648\) | \(2.4955\) | |
| 47040.hc2 | 47040dl3 | \([0, 1, 0, -1152545, -299645025]\) | \(43919722445768/15380859375\) | \(59295096000000000000\) | \([4]\) | \(1179648\) | \(2.4955\) | |
| 47040.hc3 | 47040dl2 | \([0, 1, 0, -1029065, -402059337]\) | \(250094631024064/62015625\) | \(29884728384000000\) | \([2, 2]\) | \(589824\) | \(2.1489\) | |
| 47040.hc4 | 47040dl1 | \([0, 1, 0, -56660, -7846350]\) | \(-2671731885376/1969120125\) | \(-14826560869512000\) | \([2]\) | \(294912\) | \(1.8023\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47040.hc have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.hc do not have complex multiplication.Modular form 47040.2.a.hc
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
